IntroductionKnown resultsMain theorem

On the minimal coloring number ofthe minimal diagram of torus links

Eri Matsudo

Nihon University

Graduate School of Integrated Basic Sciences

Joint work withK. Ichihara (Nihon Univ.) & K. Ishikawa (RIMS, Kyoto Univ.)

Waseda University, December 24, 2018

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IntroductionKnown resultsMain theorem

Z-coloring

Let L be a link, and D a diagram of L.

Z-coloring

A map γ : {arcs of D} → Z is called a Z-coloring on D if itsatisfies the condition 2γ(a) = γ(b) + γ(c) at each crossing of Dwith the over arc a and the under arcs b and c.A Z-coloring which assigns the same color to all the arcs of thediagram is called a trivial Z-coloring.L is Z-colorable if ∃ a diagram of L with a non-trivial Z-coloring.

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Let L be a Z-colorable link.

Minimal coloring number

[1] For a diagram D of L,mincolZ(D) := min{#Im(γ) | γ : non-tri. Z-coloring on D}

[2] mincolZ(L) := min{mincolZ(D) | D : a diagram of L}

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Simple Z-coloring

γ : a Z-coloring on a diagram D of a non-trivial Z-colorable link LIf ∃ d ∈ N s.t. at each crossings in D, the differences between thecolors of the over arcs and the under arcs are d or 0, then we call γa simple Z-coloring.

Theorem 1 [Ichihara-M., JKTR, 2017]

Let L be a non-splittable Z-colorable link. If there exists a simpleZ-coloring on a diagram of L, then mincolZ(L) = 4.

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Simple Z-coloring

γ : a Z-coloring on a diagram D of a non-trivial Z-colorable link LIf ∃ d ∈ N s.t. at each crossings in D, the differences between thecolors of the over arcs and the under arcs are d or 0, then we call γa simple Z-coloring.

Theorem 1 [Ichihara-M., JKTR, 2017]

Let L be a non-splittable Z-colorable link. If there exists a simpleZ-coloring on a diagram of L, then mincolZ(L) = 4.

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Theorem 2 [M., to apper JKTR, Zhang-Jin-Deng]

Any Z-colorable link has a diagram admitting a simple Z-coloring.

Colorally

L : a Z-colorable link

mincolZ(L) =

{2 (L : splittable)4 (L : non-splittable)

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Theorem 2 [M., to apper JKTR, Zhang-Jin-Deng]

Any Z-colorable link has a diagram admitting a simple Z-coloring.

Colorally

L : a Z-colorable link

mincolZ(L) =

{2 (L : splittable)4 (L : non-splittable)

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One of key moves of the proof

→ The obtained diagrams are often complicated.

Problem

mincolZ(Dm) =? for a minimal diagram Dm of a Z-colorable link.

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IntroductionKnown resultsMain theorem

One of key moves of the proof

→ The obtained diagrams are often complicated.

Problem

mincolZ(Dm) =? for a minimal diagram Dm of a Z-colorable link.

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IntroductionKnown resultsMain theorem

One of key moves of the proof

→ The obtained diagrams are often complicated.

Problem

mincolZ(Dm) =? for a minimal diagram Dm of a Z-colorable link.

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IntroductionKnown resultsMain theorem

One of key moves of the proof

→ The obtained diagrams are often complicated.

Problem

mincolZ(Dm) =? for a minimal diagram Dm of a Z-colorable link.

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Theorem 3 [Ichihara-M., Proc.Inst.Nat.Sci., Nihon Univ., 2018]

[1] For an even integer n ≥ 2, the pretzel link P (n,−n, · · · , n,−n)with at least 4 strands has a minimal diagram Dm s.t.mincolZ(Dm) = n+ 2.

[2] For an integer n ≥ 2, the pretzel link P (−n, n+ 1, n(n+ 1))has a minimal diagram Dm s.t. mincolZ(Dm) = n2 + n+ 3.

[3] For even integer n > 2 and non-zero integer p, the torus linkT (pn, n) has a minimal diagram Dm s.t. mincolZ(Dm) = 4.

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Theorem 4 [Ichihara-Ishikawa-M., In progress]

Let p, q and r be non-zero integers such that |p| ≥ q ≥ 1 andr ≥ 2. If pr or qr are even, the torus link T (pr, qr) has a minimaldiagram Dm s.t.

mincolZ(Dm) =

{4 (r : even)′′5′′ (r : odd)

Remark

A torus link T (pr, qr) is Z-colorable if and only if pr or qr are even.

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[Proof of Theorem 4 (In the case r:even)]Let D be the following minimal diagram of T (pr, qr).

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In the following, we will find a Z-coloring γ on D by assigningcolors on the arcs of D.We devide such arcs into q subfamilies x1, · · · ,xq.

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We first find a local Z-coloring γ. In the case r is even, we startwith setting γ(xi) = (γ(xi,1), γ(xi,2), · · · , γ(xi,r))= (1, 0, · · · , 0, 1) for any i.

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We can extend γ on the arcs in the regions (1) and (q + 1).

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We can extend γ on the arcs in the regions (2), (3), · · · , (q).

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Now, γ can be extended on all the arcs in the region depicted asfollows.

Since D is composed of p copies of the local diagram, it concludesthat D admits a Z-coloring with only four colors 0, 1, 2 and 3.

�

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Now, γ can be extended on all the arcs in the region depicted asfollows.

Since D is composed of p copies of the local diagram, it concludesthat D admits a Z-coloring with only four colors 0, 1, 2 and 3.

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Thank youfor your attention.

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